nach oben

Erschienen in:

2018 | OriginalPaper | Buchkapitel

1. Ornstein-Uhlenbeck Pinball and the Poincaré Inequality in a Punctured Domain

verfasst von : Emmnuel Boissard, Patrick Cattiaux, Arnaud Guillin, Laurent Miclo

Erschienen in: Séminaire de Probabilités XLIX

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

In this paper we study the Poincaré constant for the Gaussian measure restricted to \(D={\mathbb R}^d - \mathbb {B}\) where \(\mathbb {B}\) is the disjoint union of bounded open sets. We will mainly look at the case where the obstacles are Euclidean balls B(x _i, r _i) with radii r _i, or hypercubes with vertices of length 2r _i, and d ≥ 2. This will explain the asymptotic behavior of a d-dimensional Ornstein-Uhlenbeck process in the presence of obstacles with elastic normal reflections (the Ornstein-Uhlenbeck pinball).

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nächstes Kapitel A Probabilistic Look at Conservative Growth-Fragmentation Equations

Nur mit Berechtigung zugänglich

M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Books on Intermediate and Advanced Mathematics (Dover Publications Inc., New York, 1972)

S.B. Alexander, I.D. Berg, R.L. Bishop, The Riemannian obstacle problem. Illinois J. Math. 31, 167–184 (1987)MathSciNetMATH

C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, G. Scheffer, Sur les inégalités de Sobolev logarithmiques. Panoramas et Synthèses, vol. 10 (Société Mathématique de France, Paris, 2000)

A. Athreya, T. Kolba, J.C. Mattingly, Propagating Lyapunov functions to prove noise–induced stabilization. Electron. J. Probab. 17, 1–38 (2012)MathSciNetMATH

D. Bakry, F. Barthe, P. Cattiaux, A. Guillin, A simple proof of the Poincaré inequality for a large class of probability measures. Electron. J. Probab. 13, 60–66 (2008)CrossRef

D. Bakry, P. Cattiaux, A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254, 727–759 (2008)MathSciNetCrossRef

J. Birrell, D.P. Herzog, J. Wehr, Transition from ergodic to explosive behavior in a family of stochastic differential equations. Stoch. Process. Appl. 122(5), 1519–1539 (2012)MathSciNetCrossRef

S.G. Bobkov, Spectral gap and concentration for some spherically symmetric probability measures, in Geometric Aspects of Functional Analysis, Israel Seminar 2000–2001. Lecture Notes in Mathematics, vol. 1807 (Springer, Berlin, 2003), pp. 37–43

P. Cattiaux, Hypoellipticité et hypoellipticité partielle pour les diffusions avec une condition frontière. Ann. Inst. Henri Poincaré. Prob. Stat. 22, 67–112 (1986)MathSciNetMATH

10.

P. Cattiaux, Régularité au bord pour les densités et les densités conditionnelles d’une diffusion réfléchie hypoelliptique. Stochast. Stoch. Rep. 20, 309–340 (1987)CrossRef

11.

P. Cattiaux, Calcul stochastique et opérateurs dégénérés du second ordre: II Problème de Dirichlet. Bull. Sci. Math. 115, 81–122 (1991)MathSciNetMATH

12.

P. Cattiaux, A. Guillin, Deviation bounds for additive functionals of Markov processes. ESAIM Probab. Stat. 12, 12–29 (2008)MathSciNetCrossRef

13.

P. Cattiaux, A. Guillin, F.Y. Wang, L. Wu, Lyapunov conditions for super Poincaré inequalities. J. Funct. Anal. 256(6), 1821–1841 (2009)MathSciNetCrossRef

14.

P. Cattiaux, A. Guillin, P.A. Zitt, Poincaré inequalities and hitting times. Ann. Inst. Henri Poincaré. Prob. Stat. 49(1), 95–118 (2013)CrossRef

15.

P. Cattiaux, M. Fradon, A. Kulik, S. Roelly, Long time behavior of stochastic hard balls systems. Bernoulli 22(2), 681–710 (2016)MathSciNetCrossRef

16.

P. Diaconis, G. Lebeau, L. Michel, Geometric analysis for the Metropolis algorithm on Lipschitz domains. Invent. Math. 185(2), 239–281 (2011)MathSciNetCrossRef

17.

R. Douc, G. Fort, E. Moulines, P. Soulier, Practical drift conditions for subgeometric rates of convergence. Ann. Appl. Probab. 14(3), 1353–1377 (2004)MathSciNetCrossRef

18.

N. Down, S.P. Meyn, R.L. Tweedie, Exponential and uniform ergodicity of Markov processes. Ann. Prob. 23(4), 1671–1691 (1995)MathSciNetCrossRef

19.

M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes. Studies in Mathematics (De Gruyter, Berlin, 1994)

20.

L. Gatteschi, New inequalities for the zeros of confluent hypergeometric functions, in Asymptotic and Computational Analysis (Winnipeg, MB, 1989). Lecture Notes in Pure and Applied Mathematics, vol. 124 (Dekker, New York, 1990), pp. 175–192

21.

A. Göing-Jaeschke, M. Yor, A survey and some generalizations of Bessel processes. Bernoulli 9(2), 313–349 (2003)MathSciNetCrossRef

22.

P. Graczyk, T. Jakubowski, Exit times and Poisson kernels of the Ornstein-Uhlenbeck diffusion. Stoch. Models 24(2), 314–337 (2008)MathSciNetCrossRef

23.

T. Hargé, Diffraction pour l’équation de la chaleur. Duke Math. J. 73(3), 713–744 (1994)MathSciNetCrossRef

24.

N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North Holland, Amsterdam, 1981)MATH

25.

A.M. Kulik, T.D. Tymoshkevych, Lift zonoid order and functional inequalities. Teor. Imovir. Mat. Stat. 89, 75–90 (2013). Reprinted in Theory Probab. Math. Stat. 89, 83–99 (2014)

26.

E.H. Lieb, R. Seiringer, J. Yngvason, Poincaré inequalities in punctured domains. Ann. Math. 158, 1067–1080 (2003)MathSciNetCrossRef

27.

D. Revuz, M. Yor, Continuous Martingales and Brownian Motion (Springer, Berlin, 1991)CrossRef

28.

L.J. Slater, The real zeros of the confluent hypergeometric function. Proc. Camb. Philos. Soc. 52, 626–635 (1956)MathSciNetCrossRef

Titel: Ornstein-Uhlenbeck Pinball and the Poincaré Inequality in a Punctured Domain
verfasst von: Emmnuel Boissard
Patrick Cattiaux
Arnaud Guillin
Laurent Miclo
Verlag: Springer International Publishing
Buch: Séminaire de Probabilités XLIX
Print ISBN: 978-3-319-92419-9

Electronic ISBN: 978-3-319-92420-5

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-319-92420-5_1